Implications of Probabilistic Data Modeling for Rule Mining Michael Hahsler, Kurt Hornik and Thomas Reutterer Wirtschaftsuniversit¨ at Wien 29th Annual Conference of the German Classification Society (GfKl 2005) Magdeburg, March 9-11, 2005
Motivation • Mining association rules is an important technique for discovering meaningful patterns in transaction databases. – Example: diapers ⇒ beer – Applications: product assortment decisions, adapting promotional activities, personalized product recommendations, adaptive user interfaces • Current literature focuses on the properties of algorithms. • We will discuss properties of – transaction data sets and – interest measures from a probabilistic point of view. M. Hahsler, K. Hornik and T. Reutterer 2 Magdeburg, March 9-11, 2005
Outline 1. Association rules 2. Probabilistic model for transaction data 3. Simulation with R 4. Implications for confidence and lift 5. New measure: hyperlift 6. Conclusion M. Hahsler, K. Hornik and T. Reutterer 3 Magdeburg, March 9-11, 2005
Association Rules An association rule is a rule of the form X ⇒ Y , where X and Y are two disjoint sets of items (itemsets). Rule selection with threshold on interest measures: • Support: fraction of transactions containing an itemset • Confidence: probability of seeing Y under the condition that the transactions also contain X Found rules are often ranked by: • Lift: how many times more often X and Y occur together than expected if they where statistically independent M. Hahsler, K. Hornik and T. Reutterer 4 Magdeburg, March 9-11, 2005
A simple probabilistic framework for transaction data Transactions occur following a Poisson process time Tr1Tr2 Tr3 Tr4 Tr5 Trm-2 Trm-1 Trm 0 t We analyze transactions which are recorded in a fixed time interval of length t . The number of transactions m in the time interval is then poisson distributed with parameter θt : P ( M = m ) = e − θt ( θt ) m (1) m ! M. Hahsler, K. Hornik and T. Reutterer 5 Magdeburg, March 9-11, 2005
A simple probabilistic framework (cont’d) • n independent items L = { l 1 , l 2 , . . . , l n } , • with each having a fixed success probabilities to occur in a transaction given by the vector p = ( p 1 , p 2 , . . . , p n ) . Following the framework: c i , the observed number of transactions item l i is contained in, can be interpreted as a realization of a random variable C i . Under the condition of a fixed number of transactions m this random variable has a binomial distribution: � m � p c i i (1 − p i ) m − c i P ( C i = c i | M = m ) = (2) c i M. Hahsler, K. Hornik and T. Reutterer 6 Magdeburg, March 9-11, 2005
A simple probabilistic framework (cont’d) Since for a fixed time interval t the number of transactions m is not fixed, the unconditional distribution gives: ∞ � P ( C i = c i ) = P ( C i = c i | M = m ) · P ( M = m ) m = c i ∞ i (1 − p i ) m − c i e − θt ( θt ) m � m � � p c i = m ! c i m = c i (3) ∞ = e − θt ( p i θt ) c i ((1 − p ) θt ) m − c i � c i ! ( m − c i )! m = c i = e − p i θt ( p i θt ) c i c i ! which has a Poisson distribution with parameter λ i = p i θt . M. Hahsler, K. Hornik and T. Reutterer 7 Magdeburg, March 9-11, 2005
A simple probabilistic framework (cont’d) Representation of transaction data as a binary incidence matrix: items l 1 l 2 l 3 ... l n p 0.005 0.01 0.0003 ... 0.025 Tr 1 0 1 0 ... 1 Tr 2 0 1 0 ... 1 transactions Tr 3 0 1 0 ... 0 Tr 4 0 0 0 ... 0 . . . . . . . . . . . . . . . . . . Tr m-1 1 0 0 ... 1 Tr m 0 0 1 ... 1 c 99 201 7 ... 411 M. Hahsler, K. Hornik and T. Reutterer 8 Magdeburg, March 9-11, 2005
Simulation For simplicity we will assume for the following simulation that the parameters in λ are chosen from a single gamma distribution with parameters k = 0 . 75 and a = 250 . We will simulate the counts c i , for n = 200 different items over a t = 30 day period with transaction intensity θ = 300 transactions per day. > m <- rpois(1, theta * t) [1] 8885 > p <- sort(rgamma(n, shape = k, scale = a)/m, + decreasing = TRUE) Now we can simulate the transactions in the database by m Bernoulli trials for each of the n items and calculate the count vector c . > Tr <- matrix(rbinom(m * n, 1, p), ncol = n, byrow = TRUE) > c <- (apply(Tr, 2, sum)) M. Hahsler, K. Hornik and T. Reutterer 9 Magdeburg, March 9-11, 2005
Simulation (cont’d) We can directly calculate the support of each item from the transaction counts. > supp1 <- c/m > plot(supp1, type = "h", xlab = "items", + ylab = "support") M. Hahsler, K. Hornik and T. Reutterer 10 Magdeburg, March 9-11, 2005
M. Hahsler, K. Hornik and T. Reutterer 11 Magdeburg, March 9-11, 2005
Simulation (cont’d) Next, we extend the framework to the occurrences of 2 -itemsets with a symmetric n × n count matrix c2 and a support matrix ( supp2 ): > c2 <- sapply(1:n, function(i) { + apply(Tr[, i] & Tr[, 1:n], 2, sum)}) > diag(c2) <- NA > supp2 <- c2/m > persp(supp2, expand = 0.5, ticktype = "detailed", + border = 0, shade = 1, zlab = "support", + xlab = "items", ylab = "items") M. Hahsler, K. Hornik and T. Reutterer 12 Magdeburg, March 9-11, 2005
M. Hahsler, K. Hornik and T. Reutterer 13 Magdeburg, March 9-11, 2005
Implications for confidence Confidence is defined by conf( X ⇒ Y ) = supp( X + Y ) . (4) supp( X ) From our 2 -itemsets we can generate rules of the from l i ⇒ l j , where i, j = 1 , 2 , . . . , n and i � = j . We calculate confidence for the n ( n − 1) possible rules in the data set. > conf2 <- supp2/supp1 > persp(conf2, expand = 0.5, ticktype = "detailed", + border = 0, shade = 1, zlab = "confidence", + xlab = "items", ylab = "items") M. Hahsler, K. Hornik and T. Reutterer 14 Magdeburg, March 9-11, 2005
M. Hahsler, K. Hornik and T. Reutterer 15 Magdeburg, March 9-11, 2005
Implications for confidence (cont’d) • Confidence values are generally very low which reflect the fact that there are no associations in the data. • Some rules with confidence of one. However, left-hand-sides ( X ) have low support. • Confidence increases with the item in the right-hand-side Y of the rule getting more frequent. The fact that confidence systematically favors some rules makes the measure problematic when it comes to ranking rules. M. Hahsler, K. Hornik and T. Reutterer 16 Magdeburg, March 9-11, 2005
Implications for lift Typically, rules mined using minimum support (and confidence) are filtered or ordered using their lift value. The measure lift is defined as: lift( X ⇒ Y ) = conf( X ⇒ Y ) (5) supp( Y ) A lift value close to 1 indicates that the items are co-occurring in the database as expected under independence. > lift <- conf2/matrix(supp1, ncol = n, nrow = n, + byrow = TRUE) > persp(lift, expand = 0.5, ticktype = "detailed", + border = 0, shade = 1, zlab = "lift", + xlab = "items", ylab = "items") > length(which(lift > 2)) [1] 3424 M. Hahsler, K. Hornik and T. Reutterer 17 Magdeburg, March 9-11, 2005
M. Hahsler, K. Hornik and T. Reutterer 18 Magdeburg, March 9-11, 2005
Implications for lift (cont’d) To counter the problem with extremely high lift values, we discard all 2-itemsets which do not satisfy a minimum support of 0.1%. > min_supp <- 0.001 > length(lift[supp2 >= min_supp]) [1] 7096 > lift[supp2 < min_supp] <- 1 > persp(lift, expand = 0.5, ticktype = "detailed", + border = 0, shade = 1, zlab = "lift", + xlab = "items", ylab = "items") > length(which(lift > 2)) [1] 130 M. Hahsler, K. Hornik and T. Reutterer 19 Magdeburg, March 9-11, 2005
M. Hahsler, K. Hornik and T. Reutterer 20 Magdeburg, March 9-11, 2005
Implications for lift (cont’d) • Lift performs poorly to filter random noise in transaction data especially if for relatively rare items. • Lift has a tendency to produce higher values for rules with items close to minimum support. This makes using lift problematic for ranking discovered rules. M. Hahsler, K. Hornik and T. Reutterer 21 Magdeburg, March 9-11, 2005
New measure: hyperlift • The n × n co-occurrence matrix can be modeled by n 2 random variables C i,j . • The framework results in hypergeometric distributions for the C i,j s (urn model). • Using the expected value of C i,j lift can be rewritten as: lift( l i ⇒ l j ) = P ( l i + l j ) c i,j P ( l i ) P ( l j ) = (6) E [ C i,j ] • As a more conservative approach we use quantile Q δ [ C i,j ] instead of the expected value. c i,j hyperlift( l i ⇒ l j ) = Q δ [ C i,j ] . (7) M. Hahsler, K. Hornik and T. Reutterer 22 Magdeburg, March 9-11, 2005
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